**From:** Peter F. Patel-Schneider (*[email protected]*)

**Date:** 10/10/01

**Next message:**Pat Hayes: "model theory and literals"**Previous message:**Richard Fikes: "Re: Semantic Web Class"**In reply to:**Peter F. Patel-Schneider: "Re: new model theory for DAML+OIL"**Next in thread:**Pat Hayes: "Re: new model theory for DAML+OIL"**Reply:**Pat Hayes: "Re: new model theory for DAML+OIL"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ] [ attachment ]**Mail actions:**[ respond to this message ] [ mail a new topic ]

I didn't account for blank nodes in the model theory in a way that is easy to define a decent notion of entailment, so I put together a revision of the model theory. The development of interpretations and models is now similar to yours. I believe that entailment has G |= G' for G' isomorphic to G, which would have been somewhat difficult to achieve in the previous model theory. peter Alternative Model Theory for RDF and RDF Schema plus Datatypes Here is an alternative model theory for RDF and RDF Schema. The attempt here is to have a model theory for RDF and RDF Schema that can be extended to datatypes, as I have proposed them. This is a VERY terse document, missing most of what would correspond with Pat Hayes's excellent commentary on his model theory for RDF and RDF Schema. Note that this model theory has not gone through any screening. I have tried to make it as error-free as possible, but there are undoubtably errors remaining. In particular, there may be (little) pieces missing concerning some of the RDF and RDF Schema vocabulary. Note also that this is an unofficial and draft model theory. Peter F. Patel-Schneider 1/ RDF Graph Syntax I'm sticking with graphs, even though they have some problems for RDF. Triples have their own problems with respect to RDF. URI is a collection of URI names. URI may be the collection of all URI names, but this is not required. This model theory ignores all aspects of the structure of URIs. L is the collection of literals, disjoint from URI. Literals form the lexical space, in XML Schema datatype terms, not the value space! An untidy RDF graph, R, is a three-tuple (that can be considered to be a partially node labeled, directed triple-graph) < N, E, LN > where N is the set of nodes in the graph LN :(partial) N -> URI u L gives labels for nodes E <= N' x N'' x N is the set of edges in the graph where N' = { n : LN(n) is undefined or LN(n) in URI } where N'' = { n : LN(n) is defined and LN(n) in URI } This accounts for literals not being allowed as ``labels'' of edges, nor as the labels of nodes that are heads of edges, but does not account for edge ``labels'' being properties. An untidy RDF graph is ground if LN is a total function on N. A tidy RDF graph (also called an RDF graph) is an untidy RDF graph where LN is injective on URI (but not necessarily total). Tidy graphs here do not have to be tidy on literals, which is change from Pat Hayes's model theory. I have tried to keep as much of the terminology from Pat Hayes's model theory as possible. The union of two untidy RDF graphs, R = < N , E , LN > and R' = < N', E', LN' > is the untidy RDF graph R u R' = < N u N', E u E', LN u LN' > provided that LN and LN' agree where they are both defined. Note that N and N' need not be disjoint. The tidy version of an untidy RDF graph R = < N, E, LN > is the graph t(R) = < N/eLN, E/eLN, LN/eLN > where eLN is the equivalence relation induced by LN range restricted to URI i.e., n = n' iff LN(n) and LN(n') both defined, LN(n) in URI, and LN(n) = LN(n') and / is point congruence. [Is this the right terminology?] 2/ Literal Values LV is some collection of literal values. Literal values form the value space, in XML Schema datatype terms, not the lexical space! XLS : L -> powerset ( LV ), maps literals into the set of literal values that they might have. Here is the first substantive difference from Pat Hayes's model theory. The XLS mapping does not provide a definitive answer for the meaning of a literal. The reason for not pinning down the mapping for literals is to allow different mappings for different datatypes. For example, a node with literal label 05 might be mapped into the integer 5 or the string "05". 3/ Models and Entailment An interpretation I is a four-tuple < IR, IP, IEXT, IS > where IR is a non-empty set, called resources IP <= IR, called properties IEXT : IP -> powerset ( IR x (IR u LV) ) IS : URI -> IR An interpretation I = < IR, IP, IEXT, IS> is a model for an untidy RDF graph R = < N, E, LN > if there is a mapping M : N -> IR u LV such that for n, s, p, o in N 1. if LN(n) defined and LN(n) in URI then M(n) = IS(LN(n)) 2. if LN(n) defined and LN(n) in L then M(n) in XLS(LN(n)) 3. if <s,p,o> is in E then M(p) in IP and <M(s),M(o)> in IEXT(M(p)) This works for both ground and non-ground untidy RDF graphs. One minor difference between this model theory and Pat Hayes's is that unnamed nodes can denote resources or literal values, unless they appear in the subject position of an edge. This could easily be changed to require that unnamed nodes only denote resources. An untidy RDF graph R entails another untidy RDF graph R' (R |= R') iff R u R' is defined and every model for R is also a model for R u R'. This definition of entailment admits the possiblity that the two graphs share nodes. Will this have interesting consequences? 4/ Core RDF By core RDF I mean RDF without reification or containers. A core RDF graph is an untidy RDF graph that contains nodes with the following labels: rdf:type rdf:Property and an edge <t,t,P> where t is a node with label rdf:type and P is a node with label rdf:Property Throughout this development untidy graphs are allowed. Theorems about entailment will probably have to use tidy graphs, however. A core RDF model for a core RDF graph R is a model I for R with the following extra conditions 1. x in IP iff <x,IS(rdf:Property)> in IEXT(IS(rdf:type)) 2. IEXT(IS(rdf:type)) <= IR x IR A core RDF graph R RDF-entails another core RDF graph R' iff R u R' is defined and every core RDF model for R is also a core RDF model for R u R'. 5/ RDFS A core RDFS graph is a core RDF graph that contains nodes with the following labels: rdf:type [redundant from RDF] rdf:Property [redundant from RDF] rdfs:Resource rdfs:Class rdfs:subClassOf rdfs:subPropertyOf rdfs:seeAlso rdfs:isDefinedBy rdfs:ConstraintResource rdfs:ConstraintProperty rdfs:range rdfs:domain rdfs:label rdfs:comment rdfs:Literal and the following edges (actually edges between nodes with the following labels): <rdfs:Resource, rdf:type, rdfs:Class> <rdf:Property, rdf:type, rdfs:Class> <rdfs:Class, rdf:type, rdfs:Class> [redundant] <rdfs:Literal, rdf:type, rdfs:Class> <rdf:type, rdf:type, rdf:Property> [redundant from RDF] <rdfs:subClassOf, rdf:type, rdf:Property> <rdfs:subPropertyOf, rdf:type, rdf:Property> <rdfs:seeAlso, rdf:type, rdf:Property> <rdfs:isDefinedBy, rdf:type, rdf:Property> [redundant] <rdfs:range, rdf:type, rdfs:ConstraintProperty> <rdfs:domain, rdf:type, rdfs:ConstraintProperty> <rdfs:Class, rdfs:subClassOf, rdfs:Resource> <rdfs:ConstraintResource, rdfs:subClassOf, rdfs:Resource> <rdfs:ConstraintProperty, rdfs:subClassOf, rdfs:Resource> [redundant] <rdfs:ConstraintProperty, rdfs:subClassOf, rdfs:ConstraintResource> <rdfs:isDefinedBy, rdfs:subPropertyOf, rdfs:seeAlso> <rdf:type, rdfs:range, rdfs:Class> <rdfs:subClassOf, rdfs:domain, rdfs:Class> <rdfs:subClassOf, rdfs:range, rdfs:Class> <rdfs:subPropertyOf, rdfs:domain, rdf:Property> <rdfs:subPropertyOf, rdfs:range, rdf:Property> <rdfs:seeAlso, rdfs:range, rdfs:Resource> <rdfs:isDefinedBy, rdfs:range, rdfs:Resource> [redundant] <rdfs:range, rdfs:domain, rdf:Property> <rdfs:range, rdfs:range, rdfs:Class> <rdfs:domain, rdfs:domain, rdf:Property> <rdfs:domain, rdfs:range, rdfs:Class> <rdfs:label, rdfs:domain, rdfs:Resource> [redundant] <rdfs:label, rdfs:range, rdfs:Literal> <rdfs:comment, rdfs:domain, rdfs:Resource> [redundant] <rdfs:comment, rdfs:range, rdfs:Literal> A core RDFS model for a core RDFS graph R is a core RDF model I for R with the following extra conditions: x in IR iff <x,IS(rdfs:Resource)> in IEXT(IS(rdf:type)) x in IP iff <x,IS(rdf:Property)> in IEXT(IS(rdf:type)) [redundant from RDF] if <x,y> in IEXT(IS(rdf:type)) and <y,z> in IEXT(IS(rdfs:subClassOf)) then <x,z> in IEXT(IS(rdf:type)) [2.3.2] if <x,y> in IEXT(IS(rdfs:subClassOf)) and <y,z> in IEXT(IS(rdfs:subClassOf)) then <x,z> in IEXT(IS(rdfs:subClassOf)) [2.3.2] if <x,y> in IEXT(r) and <r,s> in IEXT(IS(rdfs:subPropertyOf)) then <x,y> in IEXT(s) [2.3.3] if <x,y> in IEXT(IS(rdfs:subPropertyOf)) and <y,z> in IEXT(IS(rdfs:subPropertyOf)) then <x,z> in IEXT(IS(rdfs:subPropertyOf) [2.3.3?] x in IP and <x,IS(rdfs:ConstraintResource)> in IEXT(IS(rdf:type)) iff <x,IS(rdfs:ConstraintProperty)> in IEXT(IS(rdf:type)) [3.1.2] for y in IR, if <x,y> in IEXT(p) and <p,c> in IEXT(IS(rdfs:range)) then <y,c> in IEXT(IS(rdf:type)) [3.1.3] if <x,y> in IEXT(p) and <p,IS(rdfs:Literal)> in IEXT(IS(rdfs:range)) then y in LV Yes, this last is a special case for rdfs:Literal, but so what! if <x,y> in IEXT(p) and <p,c> in IEXT(IS(rdfs:domain)) then <x,c> in IEXT(IS(rdf:type)) [3.1.4] A core RDFS graph R RDFS-entails another core RDFS graph R' iff R u R' is defined and every core RDFS model for R is also a core RDFS model for R u R'. 6/ Datatypes (general version) Datatypes add extra structure to literals and literal values. A datatype theory is a four-tuple <LV,DT,DTC,DTS> where LV is a collection of literal values DT is a collection URIs that are also datatypes DTC : DT -> powerset ( LV ) DTS : DT -> ( L -> LV ), with DTS(d) potentially partial and DTS(d)(L) <= DTC(d) for all d DTC maps a datatype to its extension (or value space). DTS maps a datatype to a partial map from literals (or lexical space) to literal values (or value space). Each datatype provides at most one literal value for each literal via the DTS mapping. Given a datatype theory <LV,DT,DTC,DTS> define XLS(l) = { lv in LV : for some d in DT with DT(d) defined on l lv = DT(d)(l) } Given a datatype theory <LV,DT,DTC,DTS> a datatype RDFS model for a core RDFS graph R is a core RDFS model I for R, with the following extra conditions: if <s,p,l> is in E with LN(l) in L and <M(p),c> in IEXT(IS(rdfs:range)) then for any node nc with LN(nc) in DT and M(nc) = c M(l) = DTS(LN(nc)) for v in LV, if <x,v> in IEXT(p) and <p,c> in IEXT(IS(rdfs:range)) then for any node nc with LN(nc) in DT and M(nc) = c v in DTC(LN(nc)) These conditions are rather complicated for semantic conditions, so some explanation is in orer. The first condition says that literals (n) that are objects of statements must denote according to any datatype range for the predicate (M(p)) of the statement. The second condition says that literals values (y) that are in relationships must belong to the value space (DTC(LN(c))) of any range of the relationship. A core RDFS graph R datatype-entails another core RDFS graph R' iff R u R' is defined and every datatype RDFS model for R is also a datatype RDFS model for R u R'. 7/ Datatypes (for XML Schema datatypes) A XML Schema datatype theory is a datatype theory <LV,DT,DTC,DTS> where LV contains the value spaces of the primitive XML Schema datatypes DT is the subset of URIs consisting of (built-in or all) XML Schema datatypes, distinguished either by their special names (e.g., xsd:integer) or by ``following'' them and finding an XML Schema datatype expression DTC maps each d in DT to its value space DTS maps each d in DT to its map from lexical space to value space Given an XML Schema datatype theory T an XML Schema datatype RDFS model for a core RDFS graph R is a datatype RDFS model for R over T. A core RDFS graph R xsd-entails another core RDFS graph R' iff R u R' is defined and every XML Schema datatype RDFS model for R is also an XML Schema datatype RDFS model for R u R'.

**Next message:**Pat Hayes: "model theory and literals"**Previous message:**Richard Fikes: "Re: Semantic Web Class"**In reply to:**Peter F. Patel-Schneider: "Re: new model theory for DAML+OIL"**Next in thread:**Pat Hayes: "Re: new model theory for DAML+OIL"**Reply:**Pat Hayes: "Re: new model theory for DAML+OIL"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ] [ attachment ]**Mail actions:**[ respond to this message ] [ mail a new topic ]

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